Documentation

Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

The Special Linear group $SL(n, R)$ #

This file defines the elements of the Special Linear group SpecialLinearGroup n R, consisting of all square R-matrices with determinant 1 on the fintype n by n. In addition, we define the group structure on SpecialLinearGroup n R and the embedding into the general linear group GeneralLinearGroup R (n → R).

Main definitions #

Notation #

For m : ℕ, we introduce the notation SL(m,R) for the special linear group on the fintype n = Fin m, in the locale MatrixGroups.

Implementation notes #

The inverse operation in the SpecialLinearGroup is defined to be the adjugate matrix, so that SpecialLinearGroup n R has a group structure for all CommRing R.

We define the elements of SpecialLinearGroup to be matrices, since we need to compute their determinant. This is in contrast with GeneralLinearGroup R M, which consists of invertible R-linear maps on M.

We provide Matrix.SpecialLinearGroup.hasCoeToFun for convenience, but do not state any lemmas about it, and use Matrix.SpecialLinearGroup.coeFn_eq_coe to eliminate it in favor of a regular coercion.

References #

Tags #

matrix group, group, matrix inverse

def Matrix.SpecialLinearGroup (n : Type u) [DecidableEq n] [Fintype n] (R : Type v) [CommRing R] :
Type (max 0 u v)

SpecialLinearGroup n R is the group of n by n R-matrices with determinant equal to 1.

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    SpecialLinearGroup n R is the group of n by n R-matrices with determinant equal to 1.

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      This instance is here for convenience, but is literally the same as the coercion from hasCoeToMatrix.

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      theorem Matrix.SpecialLinearGroup.ext_iff {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) (B : Matrix.SpecialLinearGroup n R) :
      A = B ∀ (i j : n), A i j = B i j
      theorem Matrix.SpecialLinearGroup.ext {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) (B : Matrix.SpecialLinearGroup n R) :
      (∀ (i j : n), A i j = B i j)A = B
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      • Matrix.SpecialLinearGroup.hasOne = { one := 1, }
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      • Matrix.SpecialLinearGroup.instInhabited = { default := 1 }

      The transpose of a matrix in SL(n, R)

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      • A.transpose = (A).transpose,
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        The transpose of a matrix in SL(n, R)

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          theorem Matrix.SpecialLinearGroup.coe_mk {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R) (h : A.det = 1) :
          A, h = A
          @[simp]
          theorem Matrix.SpecialLinearGroup.coe_inv {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) :
          A⁻¹ = (A).adjugate
          @[simp]
          theorem Matrix.SpecialLinearGroup.coe_mul {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) (B : Matrix.SpecialLinearGroup n R) :
          (A * B) = A * B
          @[simp]
          theorem Matrix.SpecialLinearGroup.coe_one {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] :
          1 = 1
          @[simp]
          theorem Matrix.SpecialLinearGroup.det_coe {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) :
          (A).det = 1
          @[simp]
          theorem Matrix.SpecialLinearGroup.coe_pow {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) (m : ) :
          (A ^ m) = A ^ m
          @[simp]
          theorem Matrix.SpecialLinearGroup.coe_transpose {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) :
          A.transpose = (A).transpose
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          • Matrix.SpecialLinearGroup.instGroup = let __src := Matrix.SpecialLinearGroup.monoid; let __src_1 := Matrix.SpecialLinearGroup.hasInv; Group.mk

          A version of Matrix.toLin' A that produces linear equivalences.

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            theorem Matrix.SpecialLinearGroup.toLin'_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) (v : nR) :
            (Matrix.SpecialLinearGroup.toLin' A) v = (Matrix.toLin' A) v
            theorem Matrix.SpecialLinearGroup.toLin'_to_linearMap {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) :
            (Matrix.SpecialLinearGroup.toLin' A) = Matrix.toLin' A
            theorem Matrix.SpecialLinearGroup.toLin'_symm_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) (v : nR) :
            (Matrix.SpecialLinearGroup.toLin' A).symm v = (Matrix.toLin' A⁻¹) v
            theorem Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) :
            (Matrix.SpecialLinearGroup.toLin' A).symm = Matrix.toLin' A⁻¹
            theorem Matrix.SpecialLinearGroup.toLin'_injective {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] :
            Function.Injective Matrix.SpecialLinearGroup.toLin'

            toGL is the map from the special linear group to the general linear group

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              theorem Matrix.SpecialLinearGroup.coe_toGL {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup n R) :
              (Matrix.SpecialLinearGroup.toGL A) = (Matrix.SpecialLinearGroup.toLin' A)
              @[simp]
              theorem Matrix.SpecialLinearGroup.map_apply_coe {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : Matrix.SpecialLinearGroup n R) :
              ((Matrix.SpecialLinearGroup.map f) g) = f.mapMatrix g

              A ring homomorphism from R to S induces a group homomorphism from SpecialLinearGroup n R to SpecialLinearGroup n S.

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                theorem Matrix.SpecialLinearGroup.scalar_eq_coe_self_center {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : (Subgroup.center (Matrix.SpecialLinearGroup n R))) (i : n) :
                (Matrix.scalar n) (A i i) = A

                The center of a special linear group of degree n is the subgroup of scalar matrices, for which the scalars are the n-th roots of unity.

                An equivalence of groups, from the center of the special linear group to the roots of unity.

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                  An equivalence of groups, from the center of the special linear group to the roots of unity.

                  See also center_equiv_rootsOfUnity'.

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                    Coercion of SL n to SL n R for a commutative ring R.

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                    Formal operation of negation on special linear group on even cardinality n given by negating each element.

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                    @[simp]
                    theorem Matrix.SpecialLinearGroup.coe_neg {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [Fact (Even (Fintype.card n))] (g : Matrix.SpecialLinearGroup n R) :
                    (-g) = -g
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                    theorem Matrix.SpecialLinearGroup.SL2_inv_expl_det {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup (Fin 2) R) :
                    Matrix.det ![![A 1 1, -A 0 1], ![-A 1 0, A 0 0]] = 1
                    theorem Matrix.SpecialLinearGroup.SL2_inv_expl {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup (Fin 2) R) :
                    A⁻¹ = ![![A 1 1, -A 0 1], ![-A 1 0, A 0 0]],
                    theorem Matrix.SpecialLinearGroup.fin_two_induction {R : Type v} [CommRing R] (P : Matrix.SpecialLinearGroup (Fin 2) RProp) (h : ∀ (a b c d : R) (hdet : a * d - b * c = 1), P Matrix.of ![![a, b], ![c, d]], ) (g : Matrix.SpecialLinearGroup (Fin 2) R) :
                    P g
                    theorem Matrix.SpecialLinearGroup.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero {R : Type u_2} [Field R] (g : Matrix.SpecialLinearGroup (Fin 2) R) (hg : g 1 0 = 0) :
                    ∃ (a : R) (b : R) (h : a 0), g = Matrix.of ![![a, b], ![0, a⁻¹]],
                    theorem Matrix.SpecialLinearGroup.isCoprime_row {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup (Fin 2) R) (i : Fin 2) :
                    IsCoprime (A i 0) (A i 1)
                    theorem Matrix.SpecialLinearGroup.isCoprime_col {R : Type v} [CommRing R] (A : Matrix.SpecialLinearGroup (Fin 2) R) (j : Fin 2) :
                    IsCoprime (A 0 j) (A 1 j)
                    theorem IsCoprime.exists_SL2_col {R : Type u_1} [CommRing R] {a : R} {b : R} (hab : IsCoprime a b) (j : Fin 2) :
                    ∃ (g : Matrix.SpecialLinearGroup (Fin 2) R), g 0 j = a g 1 j = b

                    Given any pair of coprime elements of R, there exists a matrix in SL(2, R) having those entries as its left or right column.

                    theorem IsCoprime.exists_SL2_row {R : Type u_1} [CommRing R] {a : R} {b : R} (hab : IsCoprime a b) (i : Fin 2) :
                    ∃ (g : Matrix.SpecialLinearGroup (Fin 2) R), g i 0 = a g i 1 = b

                    Given any pair of coprime elements of R, there exists a matrix in SL(2, R) having those entries as its top or bottom row.

                    theorem IsCoprime.vecMulSL {R : Type u_1} [CommRing R] {v : Fin 2R} (hab : IsCoprime (v 0) (v 1)) (A : Matrix.SpecialLinearGroup (Fin 2) R) :
                    IsCoprime (Matrix.vecMul v (A) 0) (Matrix.vecMul v (A) 1)

                    A vector with coprime entries, right-multiplied by a matrix in SL(2, R), has coprime entries.

                    theorem IsCoprime.mulVecSL {R : Type u_1} [CommRing R] {v : Fin 2R} (hab : IsCoprime (v 0) (v 1)) (A : Matrix.SpecialLinearGroup (Fin 2) R) :
                    IsCoprime ((A).mulVec v 0) ((A).mulVec v 1)

                    A vector with coprime entries, left-multiplied by a matrix in SL(2, R), has coprime entries.

                    The matrix S = [[0, -1], [1, 0]] as an element of SL(2, ℤ).

                    This element acts naturally on the Euclidean plane as a rotation about the origin by π / 2.

                    This element also acts naturally on the hyperbolic plane as rotation about i by π. It represents the Mobiüs transformation z ↦ -1/z and is an involutive elliptic isometry.

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                      The matrix T = [[1, 1], [0, 1]] as an element of SL(2, ℤ)

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                        theorem ModularGroup.coe_S :
                        ModularGroup.S = Matrix.of ![![0, -1], ![1, 0]]
                        theorem ModularGroup.coe_T :
                        ModularGroup.T = Matrix.of ![![1, 1], ![0, 1]]
                        theorem ModularGroup.coe_T_inv :
                        ModularGroup.T⁻¹ = Matrix.of ![![1, -1], ![0, 1]]
                        theorem ModularGroup.coe_T_zpow (n : ) :
                        (ModularGroup.T ^ n) = Matrix.of ![![1, n], ![0, 1]]